L(s) = 1 | + 5.27·2-s + 19.8·4-s − 10.5·5-s + 7·7-s + 62.3·8-s − 55.6·10-s − 34.7·11-s − 37.2·13-s + 36.9·14-s + 170.·16-s + 10.5·17-s − 58.5·19-s − 209.·20-s − 183.·22-s + 125.·23-s − 13.7·25-s − 196.·26-s + 138.·28-s + 35.4·29-s + 291.·31-s + 399.·32-s + 55.6·34-s − 73.8·35-s − 259.·37-s − 309.·38-s − 658.·40-s + 338.·41-s + ⋯ |
L(s) = 1 | + 1.86·2-s + 2.47·4-s − 0.943·5-s + 0.377·7-s + 2.75·8-s − 1.75·10-s − 0.952·11-s − 0.795·13-s + 0.704·14-s + 2.66·16-s + 0.150·17-s − 0.707·19-s − 2.33·20-s − 1.77·22-s + 1.13·23-s − 0.109·25-s − 1.48·26-s + 0.936·28-s + 0.226·29-s + 1.69·31-s + 2.20·32-s + 0.280·34-s − 0.356·35-s − 1.15·37-s − 1.31·38-s − 2.60·40-s + 1.28·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.405886939\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.405886939\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
good | 2 | \( 1 - 5.27T + 8T^{2} \) |
| 5 | \( 1 + 10.5T + 125T^{2} \) |
| 11 | \( 1 + 34.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 37.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 10.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 58.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 35.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 291.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 259.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 338.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 6.80T + 7.95e4T^{2} \) |
| 47 | \( 1 + 250.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 536.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 35.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 57.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 481.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 363.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 581.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 693.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 353.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.44e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.48289404713994943445822704582, −13.32097880730821409169839349298, −12.38185296883303587616969433143, −11.54804003470768597486730632527, −10.49999815706740550922325046677, −8.023929318689198866989727594526, −6.91596543508203836799506883912, −5.29172520676447856530071674478, −4.26703506910866712419780518048, −2.72503130760728466642393564230,
2.72503130760728466642393564230, 4.26703506910866712419780518048, 5.29172520676447856530071674478, 6.91596543508203836799506883912, 8.023929318689198866989727594526, 10.49999815706740550922325046677, 11.54804003470768597486730632527, 12.38185296883303587616969433143, 13.32097880730821409169839349298, 14.48289404713994943445822704582